Echoes from bounded universes

We construct a general class of modified Ellis wormholes, where one asymptotic Minkowski region is replaced by a bounded 2-sphere core, characterized by asymptotic finite areal radius. We pursue an in-depth analysis of the resulting geometry, outlining that geodesic completeness is guaranteed also when the radial function asymptotically shrinks to zero. Then, we study the evolution of scalar perturbations, bringing out how these geometric configurations can in principle affect the time-domain profiles of quasinormal modes, pointing out the distinctive features with respect to other black holes or wormholes geometries.

💡 Research Summary

The paper introduces a novel class of asymmetric wormhole‑like spacetimes obtained by modifying the classic Ellis wormhole (EWH). In the standard EWH the line element is
ds² = –dt² + dx² + r²(x)(dθ² + sin²θ dφ²)
with the areal radius r(x)=√(x² + a²), where a is the throat radius and the coordinate x runs from –∞ to +∞, giving two asymptotically flat regions. The authors keep the exterior region (x ≥ 0) unchanged but replace the interior region (x < 0) by a bounded “bubble” whose areal radius approaches a finite constant R (or zero) as x → –∞. The modified radius is defined as

r²(x) = x² + a² – (x² + a² – R²) tanh²(c x²) , x < 0,

where c (>0) controls how quickly the radius departs from the parabolic EWH behaviour and settles to its asymptotic value. For c = 0 the model reduces to the ordinary EWH; for c > 0 the interior is smoothly capped.

Geometrical properties
The throat at x = 0 remains a regular minimum (r′(0)=0, r″(0)>0). In the interior a function M(x)=1−2c(x²+a²−R²) tanh(c x²) vanishes at some x_m < 0, producing a local maximum of r(x). This creates a “bubble” shape that can be visualised through embedding diagrams. When R>0 the curvature invariants tend to the values of a 2‑sphere of radius R (e.g. Ricci scalar → 2/R²), so the interior is regular. When R=0 the invariants diverge as x → –∞, but the authors demonstrate that all geodesics can be extended indefinitely; the divergence does not correspond to a physical singularity.

Matter content and energy conditions
Using an orthonormal tetrad, the Einstein tensor components are expressed in terms of the radial function and its derivatives. Assuming a perfect‑fluid‑like source T̂_a^b = diag(ρ, –τ, p, p), one finds

ρ = (1 – r′² – 2r r″)/(8π r²),
τ = (1 – r′²)/(8π r²),
p = r″/(8π r).

At the throat the energy density ρ is negative and the null, weak, strong and dominant energy conditions are all violated, exactly as in the standard Ellis wormhole. In the interior, if R>0, r′ and r″ tend to zero, so τ and ρ approach finite positive values while the lateral pressure p vanishes. If R=0, all components blow up because r→0 faster than its derivatives, yet the spacetime remains geodesically complete.

Geodesic analysis
From the Lagrangian L = ½ g_{μν} ẋ^μ ẋ^ν = k/2 (k = –1 for massive particles, 0 for photons) the conserved energy E and angular momentum L are obtained. The effective potential reads

V_eff = (1 – k r′²) L²/r² + k (1 – r′²).

Radial (ℓ = 0) geodesics can freely cross the throat and continue into the interior regardless of R. Non‑radial geodesics encounter the interior maximum at x_m; depending on the size of R they may be reflected (bounce) before reaching the asymptotic core. Smaller R produces a higher barrier and more frequent bounces. Photon orbits are governed by the condition r″ = 0; when R < a additional unstable photon rings can appear inside the bubble.

Scalar perturbations and quasinormal modes
A massless scalar field obeys □Φ = 0. After separation of variables Φ(t,x,θ,φ) = e^{-iωt}Y_{ℓm}(θ,φ)ψ_ℓ(x)/r(x) and introducing the tortoise coordinate x_* defined by dx_*/dx = 1/√(1 – r′²), the radial equation becomes a Schrödinger‑type wave equation

d²ψ_ℓ/dx_*² +